Periodicity and Rotation Number for Random Circle Homeomorphisms

TL;DR

This paper investigates the behavior of random circle homeomorphisms, establishing links between periodic cycles, fixed points, and the rationality of the rotation number in stochastic dynamical systems.

Contribution

It proves that a random periodic cycle implies a rational rotation number and clarifies the relationship between fixed points and the rotation number in random systems.

Findings

Existence of a random periodic cycle implies a rational rotation number.

A common fixed point does not necessarily mean the rotation number is an integer.

An integer mean rotation number indicates a fixed point with positive probability.

Abstract

We study discrete-time random dynamical systems where each fibre map is an orientation-preserving homeomorphism of the circle. We prove that the existence of a random periodic cycle with period at least two implies that the random rotation number is rational almost surely. Moreover, in clear contrast with the deterministic setting, we demonstrate that a common fixed point for the fibre maps does not imply that the random rotation number is an integer. Conversely, we show that if the mean random rotation number is an integer, then the fibre maps have a fixed point with positive probability.